The popular MO question "Famous mathematical quotes" has turned
up many examples of witty, insightful, and humorous writing by
mathematicians. Yet, with a few exceptions such as Weyl's "angel of
topology," the language used in these quotes gets the message
across without fancy metaphors or what-have-you. That's probably the
style of most mathematicians.

Occasionally, however, one is surprised by unexpectedly colorful
language in a mathematics paper. If I remember correctly, a paper of
Gerald Sacks once described a distinction as being

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56

Latest paper, my co-author put in "but we will choose a more painful way, because there is nothing like pain for feeling alive" but the referee jumped on it.
–
Will JagyApr 23 '10 at 5:09

16

Maybe I should expand the question to include colorful language cut from serious mathematics papers :)
–
John StillwellApr 23 '10 at 5:18

32

By the way, your remark reminds me of another in a similar spirit that made it into the Princeton Companion. In his article on algebraic geometry, János Kollár says of stacks: "Their study is strongly recommended to people who would have been flagellants in earlier times."
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John StillwellApr 23 '10 at 7:49

27

I was actually rather surprised recently by a referee who did not know the phrase “red herring”, and had to look it up. He insisted that we change it to something more understandable. It makes me wonder how much “colourful” language is weeded out by referees, and whether the mathematical literature is poorer for it.
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Harald Hanche-OlsenApr 24 '10 at 2:31

24

@Harald: If you intend your mathematical papers to be read by a wide range of readers, then write them in simple language, suitable for those who are relative beginners in English. I remember reading long ago some metaphoric phrase in a mathematics research paper, then imagining students all over the world getting out their English dictionaries, looking it up, and still not understanding what it meant. (I no longer remember what the phrase was, just this reaction to it.)
–
Gerald EdgarApr 24 '10 at 15:43

109 Answers
109

I once had to make the point that the theory of spectra-with-group-action which I was using was much simpler, more naive, than the sort of beautiful and elaborate equivariant stable homotopy created by Peter May and his school. In the preprint I described the latter as the "Chicago, or deep-dish" theory. I took those words out of the final version, thinking of international readers who might not get the pizza reference. (I substituted some other humorously intended words which were a gentle dig at Peter.)

Now that we have a well-defined bundle
map $TM \to T\;'M$ (the union of all
$\beta_x^{-1} \circ \alpha_x$), it is
clearly an equivalence $e_M$. The
proof that $e_N \circ f_* = f_\\#
> \circ e_M$ is left as a masochistic
exercise for the reader.

Volume 3, p. 103, indexed under "Idiot, any,"

These normalizations are usually
carried out with hardly a word of
motivation, as if they are so natural
that any idiot would immediately think
of doing them—in reality, of course,
the authors already knew what results
they wanted, since they were simply
reformulating a classical theory.

From Volume 5, p.59,

We are going to begin by deriving
certain classical PDE's which describe
important (somewhat idealized)
physical situations. The word "derive"
had better be taken with a hefty grain
of salt, however. What I have really
tried to do is give plausible reasons
why the physical situations should be
governed by those PDE's which the
physicists have agreed upon. I've
never really been able to understand
which parts of the standard
derivations are supposed to be
obvious, which are mathematically
simplifying assumptions, which steps
are supposed to correspond to
empirically discovered physical laws,
or even what all the words are
supposed to mean.

I haven't the slightest idea what any
of this means! But I'm almost certain
that it amounts to the similarity
argument we have given. Aren't you
glad that you aren't a mathematician
of the 17th century!?

It is customary for authors, at the end of an introduction, to warmly thank their spouse for having granted them the peaceful time needed to complete their work. I find that these thanks are far too universal and overly enthusiastic to be believable. Yet, I must say that in the present case even what would sound for the reader as exaggerated thanks would not truly reflect the extraordinary privileges I have enjoyed. Be jealous, reader, for I yet have to hear the words I dread the most: "Now is not the time to work".

He was inspired by the following famous UK comic: (en.wikipedia.org/wiki/The_Fat_Slags) I saw him give a talk on the subject once. When the phrase came up all the English people in the audience laughed and everyone else looked around with very confused expressions on their faces.
–
Joel FineApr 24 '10 at 8:14

3

This is more colloquial than you think! The Fat Slags are a pair of well-known cartoon characters from Viz magazine. Given that he's a Brit, it's surely a reference to them.
–
Kevin BuzzardApr 24 '10 at 8:20

Math Reviews used to be much more colorful. In the 1950s, Haefliger was working on groupoids, developing a lot of what is now fundamental in the theory of stacks. Palais reviewed a 1958 paper of Haefliger's, concluding with,

The first four chapters of the paper
are concerned with an extreme,
Bourbaki-like generalization of the
notion of foliation. After some
twenty-five pages and several hundred
preliminary definitions, the reader
finds that a foliation of $X$ is to be
an element of the zeroth cohomology
space of $X$ with coefficients in a
certain sheaf of groupoids. Holonomy,
the Reeb-Ehresmann stability theorems,
etc., are then generalized to this
setting. While such generalization has
its place and may in fact prove useful
in the future, it seems unfortunate to
the reviewer that the author has so
materially reduced the accessibility
of the results, mentioned above, of
Chapter V, by couching them in a
ponderous formalism that will
undoubtedly discourage many otherwise
interested readers.

I don't agree with this quote by Errett Bishop (a constructivist who developed real analysis along constructive lines), but I admire its brio:

Mathematics belongs to man, not to
God. We are not interested in
properties of the positive integers
that have no descriptive meaning for
finite man. When a man proves a
positive integer to exist, he should
show how to find it. If God has
mathematics of his own that needs to
be done, let him do it himself.

It's an odd spin on that famous Kronecker quote about the integers and God.

Number theorist Andrew Granville wrote a paper called "Prime number races" in which he studies the "race" between prime numbers $\equiv$ 1 (mod 4) and prime numbers $\equiv$ 3 (mod 4). The introduction is most certainly a colorful one:

There’s nothing quite like a day at the races...The quickening
of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant
speeds out into the lead (or the distress if another contestant dashes out ahead of yours),
and the accompanying fear (or hope) that the leader might change. And what if the race
is a marathon? Maybe one of the contestants will be far stronger than the others, taking
the lead and running at the head of the pack for the whole race. Or perhaps the race
will be more dramatic, with the lead changing again and again for as long as one cares
to watch.
Our race involves the odd prime numbers, separated into two teams depending on
the remainder when they are divided by 4:

According to en.wikipedia.org/wiki/Alt.tv.simpsons "The writers also use the newsgroup to test how observant the fans are. In the seventh season episode "Treehouse of Horror VI", the writer of segment Homer3, David S. Cohen, deliberately inserted a false equation into the background of one scene. The equation that appears is $1782^{12} + 1841^{12} = 1922^{12}$."
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Gerry MyersonApr 7 '11 at 7:25

I realise that workers in
category-theory are so at home in
their subject that they find it more
natural to think in category-theoretic
rather than set-theoretical terms, but
I would liken this to not needing to
hear once one has learned to compose
music.

Still, Feferman is quite mistaken, I believe. Categorists, like other mathematicians, won't hesitate to think in set-theoretic terms if that is what works best in a given situation.
–
Todd Trimble♦Dec 13 '11 at 6:41

I came across this little gem when preparing for a talk on Kakeya sets and the ball multiplier problem, found on page 437 of E. Stein's Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals:

We will use this process to generate
our monster, which will have a tiny
heart and many arms.

This is perhaps more of a silly play on words than colourful, but I still got a laugh out of it. One page 58 of Conway's 'The sensual (quadratic) form' while discussing Kneser's gluing method a sentence begins:

I have the book but I don't get the pun, and I feel the lesser for it. Could you please explain it, if not in comments or answers here then, say, in your MO "profile" autobiography field or in email to me?
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Will JagyApr 24 '10 at 4:08

2

It's likely that I have a very dry sense of humour. But, if Conway was being formal he would write "To further illuminate the utility of the gluing method,..". I can't help but feel that it is written the way it is quite deliberately.
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Robby McKilliamApr 24 '10 at 21:48

2

I think I see, and I agree that it was deliberate. I was looking for song titles that rhymed, as "Cupidity Fondue," "Venality of You," "Morality Imbue."
–
Will JagyApr 24 '10 at 22:50

Fulton and Harris's "Representation Theory" has a few examples of colourful language. Two of my favorites:

In recent work their* Lie-theoretic origins have been exploited to produce their representations, but to tell their story would go far beyond the scope of these lecture(r)s.

*: The finite Chevalley groups.

Any mathematician, stranded on a desert island with only these ideas and the definition of a particular Lie algebra $\mathfrak{g}$ such as $\mathfrak{sl}_n \mathbb{C}$, $\mathfrak{so}_n \mathbb{C}$, or $\mathfrak{sp}_n \mathbb{C}$, would in short order have a complete description of all the objects defined above in the case of $\mathfrak{g}$. We should say as well, however, that at the conclusion of this procedure we are left without one vital piece of information about the representations of $\mathfrak{g}$ ... this is, of course, a description of the multiplicities of the basic representations $\Gamma_a$. As we said, we will, in fact, describe and prove such a formula (the Weyl character formula); but it is of a much les straightforward character (our hypothetical shipwrecked mathematician would have to have what could only be described as a pretty good day to come up the idea) and will be left until later.

I like the following from the Introduction of Iwaniec-Kowalski: Analytic number theory (AMS, 2004):

Poisson summation for number theory is what a car is for people in modern communities – it transports things to other places and it takes you back home when applied next time – one cannot live without it.

This is not the only good one in that introduction, I let you find the others!

In Jacquet and Langlands' "Automorphic forms on GL(2)", page 154, they discuss a construction which uses some choices of intermediate objects -- of course the question whether the final result depends on those choices comes up ; here is how they treat it :

We prefer to pretend that
the difficulty does not exist. As a matter of fact for anyone lucky enough not to have been indoctrinated in the functorial point of view it doesn’t.

Here is a colorful rejoinder by D. Zagier (in his reprinted article on the dilogarithm) to colorful language by Ph. Elbaz-Vincent and H. Gangl:

[Ph. Elbaz-Vincent and H. Gangl] called these functions "polyanalogs," an amalgam of the words "analogue," "polylog," and "pollyanna" (an American term suggesting exaggerated or unwarranted optimism). Presumably the correct term for the case $m=2$ would then be "dianalog," which has a pleasing British flavo(u)r.

Two from Casselman's "A companion to Macdonald's book on p-adic spherical functions":

The word ‘´epingler’ means ‘to pin’,
and the image that comes to mind most
appropriately is that of a mounted
butterfly specimen. [Kottwitz:1984]
uses ‘splitting’ for what most call
‘´epinglage’, but this is not
compatible with the common use of
‘deploiement’, the usual French term
for ‘splitting’.) Ian Macdonald, among
others, has suggested that retaining
the French word ´epinglage in these
notes is a mistake, and that it should
be replaced by the usual translation
‘pinning.’ This criticism is quite
reasonable, but I rejected it as
leading to noncolloquial English. The
words ‘pinning’ as noun and ‘pinned’
as adjective are commonly used only to
refer to an item of clothing worn by
infants, and it just didn’t sound
right.

and

These phenomena are part of what
Langlands calls endoscopy, a word that
might be roughly justified by saying
that endoscopy is concerned with some
fine aspects of the structure of
harmonic analysis on a reductive
p-adic group. Langlands attributes the
term to Avner Ash, praising his
classical knowledge, but I was pleased
to find recently the following
quotation that shows a more vulgar
intrusion of endoscopy into the modern
world:

Jeeves: “ . . . I had no need of the
endoscope.”

Bertie: “The what?”

Jeeves: “Endoscope, sir. An instrument
which enables one to peer into the . .
. interior and discern the core.”

From Chapter 12 of Jeeves and the
feudal spirit by P. G. Wodehouse.

This discussion is about distingishing
fae jewlry from real. Since the
endoscope also has medical uses, one
could imagine an even more vulgar
usage.

He has modified the notes several times so these might not be there anymore, but I have the older copies =)

My girlfriend is a surgeon and once a month our copy of "Endoscopy" drops through the post box. I tried to out-do her recently by sitting on the sofa reading a paper of Waldspurger about "twisted endoscopy" and she suggested he was doing it wrong.
–
Kevin BuzzardApr 24 '10 at 8:22

2

You made the effort, that's what counts in the end.
–
Will JagyApr 24 '10 at 19:05

The intuition that the set of all subsets of a finite set is finite -- or
more generally, that if $A$ and $B$ are finite sets,
then so is the set $B^A$ of all functions from $A$ to $B$ -- is
a questionable intuition.
Let $A$ be the set of some $5000$ spaces for symbols
on a blank sheet of typewriter paper,
and let $B$ be the set of some $80$ symbols of a typewriter;
then perhaps $B^A$ is infinite.
Perhaps it is even incorrect to think of $B^A$ as being a set.
To do so is to postulate an entity,
the set of all possible typewritten pages,
and then to ascribe some kind of reality to this entity -- for
example,
by asserting that one can in principle survey each possible typewritten page.
But perhaps it simply is not so.
Perhaps there is no such number as $80^{5000}$;
perhaps it is always possible to write a new and different page.
Many ordinary activities are built up in a similar way from
a rather small set of symbols or actions.
Perhaps infinity is not far off in space or time or thought;
perhaps it is while engaged in an ordinary activity -- writing a page,
getting a child ready for school,
talking with someone,
teaching a class,
making love -- that we are immersed in infinity.

Having just noticed this, I am rather disturbed by the thought that out there, somewhere, someone is looking into another person's eyes and asking "do you want to immerse yourself in infinity?"
–
Yemon ChoiJun 14 '11 at 21:40

2

Or even using it as a line in a bar, heaven forfend...
–
Yemon ChoiJun 14 '11 at 21:41

@Yemon: It is told of Pontrjagin that his students gradually realised he had already solved the suggested problem , and this was very offputting. The image of "line manager" is false. A supervisor can suggest a good area in which the student might make some progress, and also to show by example how to cope with failure. "In research, the secret of success is the successful management of failure!" Also, one key question is after failure:"Why did I think this might be a good idea?" Others are: "What are the fall back positions? What are the fall forward positions?" How to manage risk?
–
Ronnie BrownNov 4 '12 at 11:00

This quote is taken from the paper "How to write a proof" by Leslie Lamport. The paper is about a system to write mathematical proofs in a more formal way. (Of course I do not share the opinion expressed in this paragraphs.)

I am rather fond of Sylvester's "Aspiring to these wide generalizations, the analysis of quadratic functions soars to a pitch from whence it may look proudly down on the feeble and vain attempts of geometry proper to rise to its level or to emulate it in its flights." (1850)

I just came across a paper of Waldhausen (On Irreducible 3-manifolds Which are Sufficiently Large) where he says "Frequently, a proof involves a sequence of constructions, each of which in turn involves alterations of some things. To avoid an orgy of notation in such cases, we often denote the altered things by the old symbols."

In the huge and austere book "Groupes algébriques" by M. Demazure and P. Gabriel we find in the last pages a "Dictionaire "Fonctoriel"", a dictionary of terms related to category theory where they have:

André Weil uses some very colourful language in the introduction of his 1946 book Foundations of Algebraic Geometry. I recommend any mathematician to read it. Here are some excerpts:

"As in other kinds of war, so in this bloodless battle with an ever retreating foe which it is our good luck to be waging, it is possible for the advancing army to outrun its services of supply and incur disaster unless it waits for the quartermaster to perform his inglorious but indispensable task."

"Of course every mathematician has a right to his own language---at the risk of not being understood; and the use sometimes made of this right by our contemporaries almost suggests that the same fate is being prepared for mathematics as once befell, at Babel, another of man's great achievements."

"... however grateful we algebraic geometers should be to the modern algebraic school for lending us temporary accommodation, makeshift constructions full of rings, ideals and valuations, in which some of us feel in constant danger of getting lost, our wish and aim must be to return at the earliest possible moment to the palaces which are ours by birthright, to consolidate shaky foundations, to provide roofs where they are missing, to finish, in harmony with the portions already existing, what has been left undone."

"...it is hoped that these may be helpful to the reader, to whom the author, having acted as his pilot until this point, heartily wishes Godspeed on his sailing away from the axiomatic shore, further and further into open sea."

In "Théorie algébrique des nombres" (in french and a great book about Dedekind rings and basic number field theory btw), Samuel frequently uses "Mézalor" as a phonetic replacemecont for "Mais alors". I guess you could translate it as "Butzen" instead of "But then". I think it was just a geeky "wink wink" at other mathematicians considering how much that locution was used in "dévissage" but I liked it anyway.